Many developing economies are faced with growing populations and deteriorating natural resources. It is often argued that economic growth will arrest these trends via cleaner technology and social change accompanied by reduced birth rates. Two questions arise: (1) What general economic, demographic, and ecological conditions favor this scenario? and (2) What adjustments, technological, demographic, or ecological, are more important in realizing this scenario? I address these questions using a two-sector growth model which includes human demographics and a renewable resource base. Using powerful numerical bifurcation techniques and rescaling arguments, I obtain the following general results. If the regeneration rate of the renewable resource base is slow relative to the rate of economic growth, population overshoot and resource collapse is more likely. Demographic adjustments are more important than technological adjustments in avoiding renewable resource degradation. Several related results are presented that support these general findings.